El e c t ro n ic
Jo ur n
a l o
f
P r
o b
a b i l i t y
Vol. 11 (2006), Paper no. 49, pages 1321–1341.
Journal URL
http://www.math.washington.edu/~ejpecp/
The lower envelope of positive self-similar
Markov processes
Loic Chaumont
Laboratoire de Probabilit´es et Mod`eles Al´eatoires, Universit´e Pierre et Marie Curie
4, Place Jussieu – 75252 Paris Cedex 05 email: [email protected]
J.C. Pardo∗ [email protected]
Abstract
We establish integral tests and laws of the iterated logarithm for the lower envelope of pos-itive self-similar Markov processes at 0 and +∞. Our proofs are based on the Lamperti representation and time reversal arguments. These results extend laws of the iterated loga-rithm for Bessel processes due to Dvoretsky and Erd¨os (11), Motoo (17) and Rivero (18).
Key words: Self-similar Markov process, L´evy process, Lamperti representation, last pas-sage time, time reversal, integral test, law of the iterated logarithm.
AMS 2000 Subject Classification: Primary 60G18, 60G51, 60B10. Submitted to EJP on June 9 2006, final version accepted December 12 2006.
1
Introduction
A real self-similar Markov processX(x), starting fromxis a c`adl`ag Markov process which fulfills a scaling property, i.e., there exists a constantα >0 such that for any k >0,
kXk(x−)αt, t≥0
(d)
= (Xt(kx), t≥0). (1.1)
Self-similar processes often arise in various parts of probability theory as limit of re-scaled pro-cesses. Their properties have been studied by the early sixties under the impulse of Lamperti’s work (14). The Markov property added to self-similarity provides some interesting features as noted by Lamperti himself in (15) where the particular case of positive self-similar Markov processes is studied. These processes are involved for instance in branching processes and frag-mentation theory. In this paper, we will considerpositiveself-similar Markov processes and refer to them aspssMp. Some particularly well known examples are transient Bessel processes, stable subordinators or more generally, stable L´evy processes conditioned to stay positive.
The aim of this work is to describe the lower envelope at 0 and at +∞ of a large class of pssMp throug integral tests and laws of the iterated logarithm (LIL for short). A crucial point in our arguments is the famous Lamperti representation of self-similar IR+–valued Markov processes. This transformation enables us to construct the paths of any such process starting fromx >0, sayX(x), from those of a L´evy process. More precisely, Lamperti (15) found the representation
Xt(x)=xexpξτ(tx−α), 0≤t≤x−αI(ξ), (1.2)
underPx, forx >0, where
τt= inf{s:Is(ξ)≥t}, Is(ξ) =
Z s 0
expαξudu , I(ξ) = lim
t→+∞It(ξ),
and whereξ is a real L´evy process which is possibly killed at an independent exponential time. Note that for t < I(ξ), we have the equality τt =R0t Xs(x)−αds, so that (1.2) is invertible and
yields a one to one relation between the class of pssMp and the one of L´evy processes.
In this work, we consider pssMp’s which drift towards +∞, i.e. limt→+∞Xt(x)= +∞, a.s. and
which fulfills the Feller property on [0,∞), so that we may define the law of a pssMp, which we will callX(0), starting from 0 and with the same transition function asX(x),x >0. Bertoin and Caballero (2) and Bertoin and Yor (3) proved that the family of processes X(x) converges, as
x↓0, in the sense of finite dimensional distributions towardsX(0) if and only if the underlying L´evy processξ in the Lamperti representation is such that
(H) ξ is non lattice and 0< m(def)= E(ξ1)≤E(|ξ1|)<+∞.
As proved by Caballero and Chaumont in (5), the latter condition is also a NASC for the weak convergence of the family (X(x)), x ≥ 0 on the Skohorod space of c`adl`ag trajectories. In the same article, the authors also provided a path construction of the process X(0). The entrance law ofX(0)has been described in (2) and (3) as follows: for everyt >0 and for every measurable functionf : IR+→IR+,
EfXt(0)= 1
mE I(−ξ)
−1f(tI(−ξ)−1)
Several partial results on the lower envelope of X(0) have already been established before, the oldest of which are due to Dvoretsky and Erd¨os (11) and Motoo (17) who studied the special case of Bessel processes. More precisely, whenX(0) is a Bessel process with dimensionδ >2, we have the following integral test at 0: iff is an increasing function then
P(Xt(0) < f(t),i.o., as t→0) =
0
1 according as Z
0+
f(t)
t
δ−42
dt t
<∞
=∞ .
The time inversion property of Bessel processes, induces the same integral test for the behaviour at +∞ of X(x), x ≥ 0. The test for Bessel processes is extended in section 4 of this paper to pssMp’s such that the upper tail of the law of the exponential functionalI(−ξ) is regularly varying. Our integral test is then written in terms of the law of this exponential functional as shown in Theorem 3.
Without giving here an exhaustive list of the results which have been obtained in that direction, we may also cite Lamperti’s own work (15) in which he used his representation to describe the asymptotic behaviour of a pssMp starting from x >0 in terms of the underlying L´evy process. Some cases where the transition function of X(x) admits some special bounds have also been studied by Xiao (21).
The most recent result concerns increasing pssMp and is due to Rivero (18) who proved the following LIL: suppose that ξ is a subordinator whose Laplace exponentφis regularly varying at infinity with indexβ ∈ (0,1) and define the function ϕ(t) = φ(log|logt|)/log|logt|, t > e, then
lim inf
t↓0
Xt(0)
(tϕ(t))1/α =α
β/α(1−β)(1−β)/α and lim inf t↑+∞
Xt(0)
(tϕ(t))1/α =α
β/α(1−β)(1−β)/α, a.s.
In Section 5 of this paper, we extend Rivero’s result to pssMp’s such that the logarithm of the upper tail of the exponential functional I(−ξ) is regularly varying at +∞. In Theorem 4, we give a LIL for the process X(0) at 0 and for the processes X(x),x≥0 at +∞. Then the lower envelope has an explicit form in terms of the tail of the law of I(−ξ).
All the asymptotic results presented in Sections 4 and 5 are consequences of general integral tests which are stated and proved in Section 3 and which may actually be applied in other situations than our ’regular’ and ’logregular’ cases. If Fq denotes the tail of the law of the truncated
exponential functionalsRTˆ−q
0 exp−ξsds, ˆT−q = inf{t:ξs ≤ −q},q ≥0, then we will show that in any case, the knowledge of asymptotic behaviour ofFq suffices to describe the lower envelope
of the processesX(x),x≥0.
Section 2 is devoted to preliminary results. We give a path decomposition of the process X(0)
2
Time reversal and last passage time of
X
(0)We consider processes defined on the space Dof c`adl`ag trajectories on [0,∞), with real values. The space D is endowed with the Skorohod’s topology and P will be our reference probability measure.
In all the rest of the paper, ξ will be a L´evy process satisfying condition (H). With no loss of generality, we will also suppose that α= 1. Indeed, we see from (1.1) that if X(x), x ≥0, is a pssMp with index α > 0, then X(x)α
is a pssMp with index 1. Therefore, the integral tests and LIL established in the sequel can easily be interpreted for any α >0.
Let us define the family of positive self-similar Markov processes ˆX(x) whose Lamperti’s repre-sentation is given by
ˆ
X(x)=xexp ˆξτˆ(t/x),0≤t≤xI(ˆξ)
, x >0, (2.4)
where ˆξ =−ξ, ˆτt = inf{s:
Rs
0 exp ˆξudu≥t}, and I(ˆξ) = R∞
0 exp ˆξsds. We emphasize that the r.v. xI(ˆξ), corresponds to the first time at which the process ˆX(x) hits 0, i.e.
xI(ˆξ) = inf{t: ˆXt(x)= 0}, (2.5)
moreover, for each x >0, the process ˆX(x) hits 0 continuously, i.e. ˆX(x)(xI(ˆξ)−) = 0.
We now fix a decreasing sequence (xn),n≥1 of positive real numbers which tends to 0 and we
set
U(y) = sup{t:Xt(0) ≤y}.
The aim of this section is to establish a path decomposition of the processX(0) reversed at time
U(x1) in order to get a representation of this time in terms of the exponential functional I(ˆξ), see Corollaries 2 and 3 below.
To simplify the notations, we set Γ =XU(0)(x
1)− and we will denote byK the support of the law of Γ. We will see in Lemma 1 that actually K = [0, x1]. For any process X that we consider here, we make the convention thatX0−=X0.
Proposition 1. The law of the process Xˆ(x) is a regular version of the law of the process
ˆ
X (def)= (X((0)U(x
1)−t)−,0≤t≤U(x1)), conditionally onΓ =x,x∈K.
Proof: The result is a consequence of Nagasawa’s theory of time reversal for Markov processes. First, it follows from Lemma 2 in (3) that the resolvent operators ofX(x) and ˆX(x),x >0 are in duality with respect to the Lebesgue measure. More specifically, for everyq ≥0, and measurable functionsf, g: (0,∞)→IR+, with
Vqf(x)(def)= E
Z ∞ 0
e−qtf(Xt(x))dt
, and Vˆqf(x)(def)= E
Z ζ
0
e−qtf( ˆXt(x))dt
we have Z ∞
Conditions of Nagasawa’s theorem are satisfied as shown in (2.6) and (2.7), then it remains to apply this result to U(x1) which is a return time such that P(0 < U(x1) < ∞) = 1, and the proposition is proved.
Another way to state Proposition 1 is to say that for any x ∈ K, the returned process ( ˆX(xI(ˆξ)−t)−,0 ≤ t ≤ xI(ˆξ)), has the same law as (Xt(0),0 ≤ t < U(x1)) given Γ = x. In (3), the authors show that when the semigroup operator of X(0) is absolutely continuous with respect to the Lebesgue measure with density pt(x, y), this process is anh-process ofX(0), the
corresponding harmonic function beingh(x) =R∞
0 pt(x,1)dt.
Proof: From (2.4) and Proposition 1, the process ˆX may be described as
where ˆξ(1) (= ˆd)ξ is independent of Γ = XU(0)(x we do not change the law of ˆX if, by reconstructing it according to this decomposition, we replace ˆξ(2) by a process with the same law which is independent of [ˆξ(1),Γ
1]. Moreover, ˆξ(2) is independent of Γ2. Relation (2.8) is a consequence of the scaling property. Indeed, we have
which implies the identities in law
x−11XU(0)(x
On the other hand, we see from the definition of ˆX in Proposition 1 that
Then, we obtain (2.8) for n = 2 from this identity, (2.9) and (2.10). The proof follows by induction.
Corollary 2. With the same notations as in Corollary 1, the time U(xn) may be decomposed
To prove the second inequality in (2.12), it suffices to note that by Proposition 1 and the strong Markov property at time ˆS(xn), for anyn≥1, we have the representation
ˆ
XSˆ(xn)+t,0≤t≤U(x1)−Sˆ(xn)
=Γnexpξ
(n)
τ(n)(t/Γ
n),0≤t≤U(x1)−
ˆ
S(xn)
,
whereτ(tn) = inf{s:Rs 0 expξ
(n)
u du > t} and Γn= ˆXSˆ(xn) (see in Corollary 1) is independent of
ξ(n) which has the same law as ˆξ. It remains to note from (2.5) that U(x1)−Sˆ(xn) =U(xn) =
ΓnI(ξ
(n)
) and that Γn≤xn.
To establish our asymptotic results at +∞, we will also need to estimate the law of the time
U(x) when x is large. The same reasoning as we did for a sequence which tends to 0 can be done for a sequence which tends to +∞as we show in the following result.
Corollary 3. Let (yn) be an increasing sequence of positive real numbers which tends to +∞.
There exists some sequences(ˇξ(n)),(˜ξ(n))and(ˇΓn), such that for eachn,ξˇ(n) ( d)
= ˜ξ(n) (= ˆd)ξ,Γˇn
(d) = Γ, Γˇn and ξˇ(n) are independent; moreover the L´evy processes (ˇξ(n)) are mutually independent
and we have for all zn>0,
zn1I{Γˇn≥zn}
Z Tˇ(n)(log(yn−1/zn))
0
exp ˇξs(n)ds≤U(yn)≤ynI(˜ξ(n)), a.s. (2.13)
where Tˇz(n)= inf{t: ˇξt(n)≤z}.
Proof: Fix an integer n≥1 and define the decreasing sequence x1, . . . , xn by xn = y1, xn−1 =
y2, . . . , x1 = yn, then construct the sequences ˆξ(1), . . . ,ξˆ(n) and Γ1, . . . ,Γn from x1, . . . , xn as
in Corollary 1 and construct the sequence ξ(1), . . . , ξ(n) as in Corollary 2. Now define ˇξ(1) = ˆ
ξ(n),ξˇ(2) = ˆξ(n−1), . . . ,ξˇ(n) = ˆξ(1)and ˜ξ(1)=ξ(n),ξ˜(2) =ξ(n−1), . . . ,ξ˜(n)=ξ(1)and ˇΓ
1= Γn,Γˇ2= Γn−1, . . . ,Γˇn= Γ1. Then from (2.12), we deduce that for anyk= 2, . . . , n,
zk1I{Γˇk≥zk}
Z Tˇ(k)(log(yk−1/zk))
0
exp ˇξs(k)ds≤U(yk)≤ykI(˜ξ(k)), a.s.
Hence the whole sequences (˜ξ(n)), (ˇξ(n)) and (ˇΓ
n) are well constructed and fulfill the desired
properties.
Remark: We emphasize that ˆT(n)(log(x
n+1/Γn)) = 0, a.s. on the event Γn≤xn+1; moreover, we have Γn ≤xn, a.s., so the first inequality in (2.12) is relevant only when xn+1 < zn < xn.
Similarly, in Corollary 2, the first inequality in (2.13) is relevant only whenyn−1< zn< yn.
We end this section with the computation of the law of Γ. Recall that the upward ladder height process (σt) associated toξis the subordinator which corresponds to the right continuous
inverse of the local time at 0 of the reflected process (ξt−sups≤tξs), see (1) Chap. V for a
Lemma 1. The law ofΓ is characterized as follows:
log(x−11Γ)(=d)−UZ ,
where U andZ are independent r.v.’s, U is uniformly distributed over [0,1]and the law of Z is given by:
P(Z > u) =E(σ1)−1 Z
(u,∞)
s ν(ds), u≥0. (2.14)
In particular, for allη < x1, P(Γ> η)>0.
Proof. It is proved in (10) that under our hypothesis, (that isE(|ξˆ1|)<+∞,E(ˆξ1)<0 andξ is not arithmetic), the overshoot process ofξ converges in law, that is
ˆ
ξTˆ(x)−x−→ −UZ, in law asx tends to −∞,
and the limit law is computed in (8) in terms of the upward ladder height process (σt).
On the other hand, we proved in Corollary 1, that
x−n+11 Γn+1 = exp[ˆξT(ˆn()n)(log(x
n+1/Γn))−log(xn+1/Γn)]
(d) = x−11Γ (d)
= exp[ˆξTˆ(log(xn+1/xn)+log(x−1
1 Γ))−log(xn+1/xn)−log(x −1 1 Γ)].
Then by takingxn=e−n
2
, we deduce from these equalities that log(x−11Γ) has the same law as the limit overshoot of the process ˆξ, i.e.
ˆ
ξTˆ(x)−x−→log(x−11Γ), in law asx tends to −∞.
As a consequence of the above results we have the following identity in law:
U(x)(=d) x
x1ΓI(ˆξ),
(Γ andI(ˆξ) being independent) which has been proved in (2), Proposition 3 is the special case where the processX(0) is increasing.
3
The lower envelope
The main result of this section consists in integral tests at 0 and +∞ for the lower envelope of
X(0). When no confusion is possible, we setI (def)= I(ˆξ) = R∞
0 exp ˆξsds. This theorem means in particular that the asymptotic behaviour ofX(0) only depends on the tail behaviour of the law of I, and on that of the law of RTˆ−q
0 exp ˆξsds, with ˆTx = inf{t : ˆξt≤x}, for x ≤0. So also we set
Iq
(def) =
Z Tˆ−q
0
exp ˆξsds , F(t)
(def)
= P(I > t), Fq(t)
(def)
= P(Iq > t).
Lemma 2. Assume that there exists γ > 1 such that, lim supt→+∞F(γt)/F(t) < 1. For any
We start by stating the integral test at time 0.
Theorem 1. The lower envelope ofX(0) at0 is described as follows:
Let f be an increasing function.
Proof: Let (xn) be a decreasing sequence such that limnxn= 0. Recall the notations of Section
Since f is increasing, the following inclusions hold:
{xn≤f(U(xn))} ⊂An⊂ {xn+1≤f(U(xn))}. (3.2) With no loss of generality, we can restrict ourselves to the casef(0) = 0, so it is not difficult to check that for anyr >1,
Suppose the latter condition holds, then from (3.5), for all r > 1, P∞
is independent of Γn, and Γn is such that x−n1Γn
(d)
= x−11Γ. Moreover the r.v.’s I(n), n ≥1 are independent between themselves and identity (3.7) shows that they have the same law as Iq
defined in Lemma 2, where q = −log(1/rk). With no loss of generality, we may assume that
f(0) = 0, so that we can write Bn = {r−n ≤ fr,ε(kr−nI(n)),Γn ≥ kr−n} and from the above
arguments we deduce
P(Bn) =P(r−n≤fr,ε(kr−nIq))P(Γ≥kr−1). (3.8)
The arguments which are developed above to show (3.5) and (3.6), are also valid if we replace
I by Iq. Hence from the hypothesis, since
R
0+P(s < f(s)Iq)ds/s = +∞, then from (3.5) and (3.6) applied toIq, we have P∞n=1P(r−(n+1) ≤f(r−nIq)) =P∞n=1P(r−n ≤fr,ε(kr−nIq)) =∞,
and from (3.8) we have P
nP(Bn) = +∞. Then another application of (3.8), gives for any n
and m,
P(Bn∩Bm) ≤ P(r−n≤fr,ε(kr−nIq))P(r−m≤fr,ε(kr−mIq))
P(Bn∩Bm) ≤ P(Γ≥kr−1)−2P(Bn)P(Bm),
whereP(Γ≥kr−1)>0, from (2.14). Hence from the extension of Borel-Cantelli’s lemma given in (13),
P(lim supBn)≥P(Γ≥kr−1)2 >0. (3.9)
Then recall from Corollary 2 the inequality kr−n1I
{Γn≥kr−n}I
(n) ≤ U(r−n) which implies from
(3.2) that Bn ⊂ An, (where in the definition of An we replaced f by fr,ε). So, from (3.9),
P(lim supnAn) > 0, but since X(0) is a Feller process and since lim supnAn is a tail event, we
have P(lim supnAn) = 1. We deduce from the scaling property of X(0) and (3.1) that
P(Xt(0) ≤fr,ε(t),i.o., as t→0.) = P(Xkt(0)≤rf(t),i.o., as t→0.)
= P(Xt(0)≤k−1rf(t),i.o., as t→0.) = 1.
Sincek= 1−ε+ε/r, withr >1 and 0< ε <1 arbitrary chosen, we obtain (ii).
Now we prove the divergence part (iii). The sequences (xn) and (zn) are defined as in the proof
of (ii) above. Recall thatq = −log(1/rk) and takeδ > γe−q as in Lemma 2. With no loss of generality, we may assume that f(t)/t → 0, as t → 0. Then from the hypothesis in (iii) and Lemma 2, we have
Z 0+
Fq
(1−δ)t f(t)
dt
t =∞.
As already noticed above, this is equivalent to R+∞
1 P((1−δ)r−t ≤ f(r−tIq))dt = ∞. Since
t 7→ f(t)/t increases, R+∞
1 P((1−δ)r−t ≤f(r−tIq))dt ≤ P∞
1 P((1−δ)r−n ≤ f(r−nIq)) = ∞.
Setfr(δ)(t) = (1−δ)−1f(t/k), then
∞ X
1
P(r−n≤fr(δ)(kr−nIq)) =∞.
Similarly as in the proof of (ii), defineBn′ ={r−n≤fr(δ)(kr−nI(n)),Γn≥kr−n}. ThenBn′ ⊂An,
P∞
P(Bn′) =∞, we haveP(lim supnAn) = 1, hence from the scaling property ofX(0) and (3.1)
P(Xt(0) ≤fr(δ)(t),i.o., ast→0.) = P(Xkt(0) ≤(1−δ)−1f(t),i.o., ast→0.)
= P(Xt(0) ≤k−1(1−δ)−1f(t),i.o., ast→0.) = 1.
Sincek= 1−ε+ε/r, withr >1 and 0< ε <1 andδ > γe−q=γ/(r+ε(1−r)), by choosingr
sufficiently large andεsufficiently small, δ can be taken sufficiently small so that k−1(1−δ)−1 is arbitrary close to 1.
The divergence part of the integral test at +∞ requires the following Lemma.
Lemma 3. For any L´evy process ξ such that 0<E(ξ1)≤E(|ξ1|)<∞, and for any q ≥0,
E inft≤Tqξt
<∞,
where Tq= inf{t:ξt≥q}.
Proof. The proof bears upon a result on stochastic bounds for L´evy processes due to Doney (9) which we briefly recall. Let νn be the time at which then-th jump of ξ whose value lies in
[−1,1]c, occurs and define
In= inf
νn≤t<νn+1
ξt.
Theorem 1.1 in (9) asserts that the sequence (In) admits the representation
In=Sn(−)+ ˜ı0, n≥0,
whereS(−)is a random walk with the same distribution as (ξ(νn), n≥0) and ˜ı0 is independent of S(−). For a≥0, letσ(a) = min{n:Sn(−)> a}, then for any q≥0, we have the inequality
min
n≤σ(q+|˜ı0|)(S (−)
n + ˜ı0)≤ inf
t≤Tq
ξt. (3.10)
On the other hand, it follows from our hypothesis on ξ that 0 < E(S1(−)) ≤E(|S1(−)|) < +∞, hence from Theorem 2 of (12) and its proof, there exists a finite constantC which depends only on the law ofS(−) such that for anya≥0,
E
minn≤σ(a)S (−)
n
≤CE(σ(a))E(|S1(−)|). (3.11)
Moreover from (1.5) in (12), there are finite constants A and B depending only on the law of
S(−) such that for anya≥0
E(σ(a))≤A+Ba . (3.12) Since ˜ı0 is integrable (see (9)), the result follows from (3.10), (3.11), (3.12) and the independence between ˜ı0 and S(−).
Theorem 2. The lower envelope ofX(x) at+∞ is described as follows:
(i) If
Proof: We first consider the case wherex = 0. The proof is very similar to this of Theorem 1. We can follow the proofs of (i), (ii) and (iii) line by line, replacing the sequences xn=r−n and
zn =kr−n respectively by the sequences xn = rn and zn =krn, and replacing Corollary 2 by
Corollary 3. Then with the definition
An={There existst∈[U(rn), U(rn+1)] such that Xt(0) < f(t).},
we see that the event lim supAnbelongs to the tail sigma-field ∩tσ{Xs(0) :s≥t}which is trivial
from the representation (1.2) and the Markov property.
The only thing which has to be checked more carefully is the counterpart at +∞ of the equivalence (3.6). Indeed, since in that case R∞
1 P(rt < f(rtI)dt =
and sincef is increasing, we have
But Z ∞
1 P
(s < Iq) ds
s =E(log
+I
q).
Note that from our hypothesis on ξ, we have E( ˆT−q) <+∞, then the conclusion follows from
the inequality
E(log+Iq)≤E
sup0≤s≤Tˆ −q
ˆ
ξs
+E( ˆT−q)
and Lemma 3. This achieves the proof of the theorem forx= 0.
Now we prove (i) for any x >0. Let f be an increasing function such that R+∞
Ff(tt) dtt <
+∞. Letx >0, putSx = inf{t:Xt(0) ≥x}and denote byµxthe law of XS(0)x. From the Markov
property at timeSx, we have for allε >0,
P(Xt(0) <(1−ε)f(t−Sx),i.o., as t→+∞)
= Z
[x,∞)P
(Xt(y) <(1−ε)f(t),i.o., as t→+∞)µx(dy)
≤ P(Xt(0) <(1−ε)f(t),i.o., as t→+∞) = 0. (3.14) Ifx is an atom of µx, then the inequality (3.14) shows that
P(Xt(x) <(1−ε)f(t),i.o., as t→+∞) = 0
and the result is proved. Suppose that x is not an atom of µx. Recall from section 1 that
log(x−11Γ) is the limit in law of the overshoot process ˆξTˆz −z, as z → +∞. Moreover, it
follows from (5), Theorem 1 that XS(0)x (=d) xx1
Γ . Hence, from Lemma 1, we have for any η >0,
µx(x, x+η) >0. Then, the inequality (3.14) implies that for anyη >0, there existsy∈(x, x+η)
such thatP(Xt(y)<(1−ε)f(t),i.o., as t→+∞) = 0, for allε >0. It allows us to conclude. Parts (ii) and (iii) can be proved through the same way.
We recall that to obtain these tests for any scaling indexα > 0, it suffices to consider the pro-cess (X(0))1/α in the above theorems. The same remark holds for the results of the next sections.
4
The regular case
The first type of tail behaviour ofI that we consider is the case whereF is regularly varying at infinity, i.e.
F(t)∼λt−γL(t), t→+∞, (4.1) whereγ >0 andLis a slowly varying function at +∞. As shown in the following lemma, under this assumption, for anyq >0 the functions Fq and F are equivalent, i.e. Fq ≍F.
Lemma 4. Recall that Iq = R0T−qexp ˆξsds and Fq(t) = P(Iq > t). If (4.1) holds then for all
q >0,
(1−e−γq)F(t)≤Fq(t)≤F(t), (4.2)
Proof: Recall from Lemma 2, that if (ˆξs, s≤Tˆ−q) and ˆξ′
of the lemma. To show the first inequality, we write for allδ >0,
P(I >(1 +δ)t) ≤ P(Iq+e−qI′ ≥(1 +δ)t)
and the result follows sinceδ can be chosen arbitrary small.
The regularity of the behaviour of F allows us to drop the εof Theorems 1 and 2 in the next integral test.
Proof: First let us check that for any constant β >0:
Now it follows from Theorem 1 that if R 0+F
t f(t)
dt
t < ∞ then for all ε > 0, P(X
(0)
t <
(1−ε)f(t),i.o., ast→0) = 0. If R 0+F
t f(t)
dt
t = ∞ then from Lemma 4, for all q > 0,
R 0+Fq
t f(t)
dt
t = ∞, and it follows from Theorem 1 (ii) that for all ε > 0, P(X
(0)
t < (1 +
ε)f(t),i.o., ast→0) = 1. Then the equivalence (4.4) allows us to dropεin these implications. The test at +∞ is proven through the same way.
Remarks:
1. It is possible to obtain the divergence parts of Theorem 3 by applying parts (iii) of Theorems 1 and 2 but then, one has to assume that f(t)/t is an increasing (respectively a decreasing) function for the test at 0 (respectively at +∞), which is slightly stronger than the hypothesis on f of Theorem 3.
2. This result is due to Dvoretzky and Erd¨os (11) and Motoo (17) when X(0) is a transient Bessel process, i.e. the square root of the solution of the SDE:
Zt= 2
Z t 0
p
ZsdBs+δt , (4.5)
whereδ >2 andB is a standard Brownian motion. (Recall that whenδis an integer,X(0) =√Z
has the same law as the norm of the δ-dimensional Brownian motion.) Processes X(0) = √Z such that Z satisfies the equation (4.5) with δ >2 are the only continuous self-similar Markov process with index α = 2, which drifts towards +∞. In this particular case, thanks to the time-inversion property, i.e.:
(Xt, t >0)
(d)
= (tX1/t, t >0),
we may deduce the test at +∞from the test at 0.
3. A possible way to improve the test at ∞ in the general case (that is in the setting of Theorem 1) would be to first establish it for the Ornstein-Uhlenbeck process associated toX(0), i.e. (e−tX(0)(et), t ≥ 0), as Motoo did for Bessel processes in (17). This would allow us to consider test functions which are not necessarily increasing.
Examples:
1. Examples of such behaviours are given by transient Bessel processes raised to any power and more generally when the processξ satisfies the so called Cramer’s condition, that is,
there exists γ >0 such thatE(e−γξ1) = 1. (4.6) In that case, Rivero (19) and Maulik and Zwart (16) proved by using results of Kesten and Goldie on tails of solutions of random equations that the behavior of P(I > t) is given by
F(t)∼Ct−γ, ast→+∞, (4.7) where the constantC is explicitly computed in (18) and (16).
initial process when it starts fromx >0 and killed at its first exit time of (0,∞). Denote by (qt)
the semigroup of a stable L´evy processY with indexα∈(0,2], killed at timeR= inf{t:Yt≤0}.
The function h(x) =xα(1−ρ), where ρ =P(Y
1 ≥0), is invariant for the semi-group (qt), i.e. for
all x≥0 andt≥0,Ex(h(Yt)1I{t<R}) =h(x), (Ex denotes the law of Y +x). The L´evy process
Y conditioned to stay positive is the strong Markov process whose semigroup is
p↑t(x, dy) := h(y)
h(x)qt(x, dy), x >0, y >0, t≥0. (4.8)
We will denote this process by X(x) when it is issued from x >0. We refer to (6) for more on the definition of L´evy processes conditioned to stay positive and for a proof of the above facts. It is easy to check that the process X(x) is self-similar and drifts towards +∞. Moreover, it is proved in (6), Theorem 6 that X(x) converges weakly as x → 0 towards a non degenerated process X(0) in the Skorohod’s space, so from (5), the underlying L´evy process in the Lamperti representation ofX(x) satisfies condition (H).
We can check that the law ofX(x)belongs to the regular case by using the equality in law (2.4). Indeed, it follows from Proposition 1 and Theorem 4 in (6) that the law of the exponential functional I is given by
P(t < xαI) =x1−αρE−x( ˆYtαρ−11I{t<Rˆ}), (4.9)
where ˆY = −Y and ˆR = inf{t : ˆYt ≤ 0}. If Y (and thus X(0)) has no positive jumps, then
αρ= 1 and it follows from (4.9) and Lemma 1 in (7) that
P(t < I) =Ct−ρ. (4.10)
We conjecture that (4.10) is also valid when Y has positive jumps. We also emphasize the possibility that the underlying L´evy process in the Lamperti representation of X(x) even satisfies (4.6) withγ =ρ.
5
The log regular case
The second type of behaviour that we shall consider is when logF is regularly varying at +∞, i.e.
−logF(t)∼λtβL(t), ast→ ∞, (5.1) whereλ >0, β >0 andL is a function which varies slowly at +∞. Define the functionψ by
ψ(t)(def)= t
inf{s: 1/F(s)>|logt|}, t >0, t6= 1. (5.2)
Then the lower envelope ofX(0) may be described as follows:
(i)
lim inf
t→0
Xt(0)
ψ(t) = 1, almost surely. (5.3)
(ii) For all x≥0,
lim inf
t→+∞
Xt(x)
ψ(t) = 1, almost surely. (5.4)
Proof: We shall apply Theorem 1. We first have to check that under hypothesis (5.1), the conditions of part (iii) in Theorem 1 are satisfied. On the one hand, from (5.1) we deduce that for anyγ >1, lim supF(γt)/F(t) = 0. On the other hand, it is easy to see that both ψ(t) and
ψ(t)/tare increasing in a neighbourhood of 0. LetL be a slowly varying function such that
−logF(λ−1/βt1/βL(t))∼t , ast→+∞. (5.5) Th. 1.5.12, p.28 in (4) ensures that such a function exists and that
inf{s:−logF(s)> t} ∼λ−1/βt1/βL(t), ast→+∞. (5.6) Then we have for allk1<1 andk2 >1 and for all tsufficiently large,
k1λ−1/βt1/βL(t)≤inf{s:−logF(s)> t} ≤k2λ−1/βt1/βL(t) so that forψ defined above and for allk′2>0,
−logF
t k′ 2
k2ψ(t)
≤ −logF(k2′λ−1/β(log|logt|)1/βL(log|logt|)) (5.7)
for all tsufficiently small. But from (5.5), for allk′′
2 >1 and for alltsufficiently small,
−logF(k′2λ−1/β(log|logt|)1/βL(log|logt|))≤k2′′k′2βlog|logt|,
hence
F
t k′2 k2ψ(t)
≥(|logt|)−k′′2k′2
β .
By choosingk′
2 <1 and k2′′<(k2′)−β, we obtain the convergence of the integral Z
0+
F
t k′ 2
k2ψ(t)
dt t ,
for all k2 >1 and k2′ <1, which proves that for allε >0,
P(Xt(0)<(1 +ε)ψ(t),i.o., ast→0) = 1
from Theorem 1 (iii). The convergence part is proven through the same way so that from Threorem 1 (i), one has for allε >0,
and the conclusion follows.
Condition (5.1) implies that ψ(t) is increasing in a neighbourhood of +∞ whereas ψ(t)/t is decreasing in a neighbourhood of +∞. Hence, the proof of the result at +∞ is done through the same way as at 0, by using Theorem 2, (i) and (iii).
Example:
An example of such a behaviour is provided by the case where the process X(0) is increasing, that is when the underlying L´evy processξ is a subordinator. Then Rivero (18), see also Maulik and Zwart (16) proved that when the Laplace exponentφof ξ which is defined by
exp(−tφ(λ)) =E(exp(λξˆt)), λ >0, t≥0
is regularly varying at +∞with indexβ ∈(0,1), the upper tail of the law ofI and the asymptotic behavior ofφat +∞ are related as follows:
Proposition 2. Suppose that ξ is a subordinator whose Laplace exponent φ varies regularly at
infinity with index β∈(0,1), then
−logF(t)∼(1−β)φ←(t), as t→ ∞, where φ←(t) = inf{s >0 :s/φ(s)> t}.
Then by using an argument based on the study of the associated Ornstein-Uhlenbeck process (e−tX(0)(et), t≥0) Rivero (18) derived from Proposition 2 the following result. Define
ϕ(t) = φ(log|logt|)
log|logt| , t > e .
Corollary 4. If φ is regularly varying at infinity with indexβ ∈(0,1) then
lim inf
t↓0
X(0)
tϕ(t) = (1−β)
1−β and lim inf t↑+∞
X(0)
tϕ(t) = (1−β)
1−β, a.s.
This corollary is also a consequence of Proposition 2 and Theorem 4. To establish Corollary 4, Rivero assumed moreover that the density of the law of the exponential functional I is decreasing and bounded in a neighbourhood of +∞. This additional assumption is actually needed to establish an integral test which involves the density ofI and which implies Corollary 4.
Acknowledgment: We are much indebted to professor Zhan Shi for many fruitful discussions
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